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**URGENT: Can someone check my working on these Diff Eqs.**

Hi I have a maths exam tomorrow, and want to make sure Im doing the right thing, please bare in mind i study physics, and am not used to some of the techniques that are used in pure maths, so if the person can help using the methods i am using, would be much appreciated.

Question 1)

Solve this first order differential equation

dy/dx = x^3/y

subject to the boundary condition that y(x=0)=1

ok so what I do is rearrange to get:

1/y dy = x^3 dx

and then integrate both sides with respect to their variable to get

ln y = 1/4 x^4 + c

so then i remove the ln:

y = e^(1/4 x^4 + c)

SO

now im a little bit confused with the boundary conditions, am I right in saying that when x = 0 , y = 1??

if that is true then am I right in saying that the constant of integration must be 0 (as e^0 = 1), so the answer is:

y = e ^ (1/4 x^4)

??

Question 2)

Solve the 2nd order differential equation:

d^2y/dx^2 - 4dy/dx = 0

(no boundary conditions this time)

ok so here i used the substitution U = dy/dx and get

dU/dx - 4U = 0

and rearrange to get:

1/U dU = 4 dx

and similarly to above integrate both sides

ln U = 4x + c1

rearrange:

dy/dx = U = e ^ (4x +c1)

ok, so now im slightly confused, do I ignore the constant becasue there are no bc's in this question???? If that is right, then I just integrate again to get:

y = 4 e^(4x) + C

is that correct??